Science and Computers -- PHY307/607

Lab 9. - The Henon Map

The purpose of this lab will be to examine the behavior of the Henon Map as we vary the parameter a (the other parameter is held fixed at b=0.3). The map takes the form 

x_n+1=a-x^2_n+by_n
y_n+1=x_n
The applet plots the values of (x,y) produced by the map after many iterations. They are the blue points drawn to the screen. The current new point is drawn as the larger red circle.
  1. Fire up Netscape and link to the Henon Applet off the Labs page.
  2. Fixed point behavior is indicated by just a single dot on the screen. A 2-cycle will be two dots, a 4-cycle 4 dots etc. By varying the parameter a try to locate the value of a at which a bifurcation from fixed point to 2-cycle behavior is observed. Call this a_1 and write it down (hint: by dragging the mouse it is possible to magnify the region around any of the dots thus allowing you to distinguish one dot from two very closely separated dots). You should be able to find the value of a to 2 decimal places this way.
  3. Increase a and try to find the transition to 4-cycle behavior -- a_2 (it's around a=0.9). Again, magnification helps enormously in finding an accurate value. You need at least 2 decimal places.
  4. Find the bifurcation to 8-cycles. Write the value of a where this happens (a_3).
  5. How many points are seen at a=1.055 ?
  6. Compute the ratio 
    
(a_2-a_1)/(a_3-a_2)
  7. You computed a similar number in Lab 3 for the logistic map. It was called the Feigenbaum number and I claimed then it was universal. Is the number you compute from the Henon map (a very different and 2D map) compatible with your previous estimate ? You should find that it is indeed a universal number which characterizes the period doubling approach to chaos.
  8. Set a=1.25. Describe what you see. This is analagous to the 3-cycle that appears in the logistic map in the middle of a chaotic region. This region is called a periodic window.
  9. See if you can follow the series of bifurcations back to chaos as a is increased still further. Specifically, at what value of a does a transition to a 14-cycle occur. How many points are seen at a=1.26 ? (Optional: If you compute the ratio of changes in a you should again find the Feigenbaum number !)
  10. Set a=1.4. Focus in on the resultant fractal - the Henon attractor. You should see that what seem at first glance to be continuous curves have a Cantor set-like structure. The strange attractor will indeed have a fractional dimension!

Drawing a snowflake

Finally, I will give you the set of moves you need to convert your Sierpinski applet to one for drawing the snowflake applet. The latter needs 5 separate moves. This means that you will to define (new) values for a[0], a[1] ... a[4] and similarly for all the arrays b[], c[] etc. You will need to make a new directory (called, for example, Snowflake), copy the contents of Simple3 to this new directory. Then edit Map.java and replace the definitions of a[] etc by the following 

a[0]=0.3333;b[0]=0.0;c[0]=0.0;d[0]=0.3333;e[0]=0.0;f[0]=0.0;
a[1]=0.3333;b[1]=0.0;c[1]=0.0;d[1]=0.3333;e[1]=0.6667;f[1]=0.0;
a[2]=0.3333;b[2]=0.0;c[2]=0.0;d[2]=0.3333;e[2]=0.0;f[2]=0.6667;
a[3]=0.3333;b[3]=0.0;c[3]=0.0;d[3]=0.3333;e[3]=0.6667;f[3]=0.6667;
a[4]=0.3333;b[4]=0.0;c[4]=0.0;d[4]=0.3333;e[4]=0.3333;f[4]=0.3333;
You will also need to change the value of the parameter K to 5 and recompile the code.
To get credit for this lab you need to
  1. Email me the url to your snowflake applet.
  2. Hand in written answers to the questions posed in the lab.
The deadline to get these things to me is next Thursday (November 4).
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This page maintained by Simon Catterall, last updated 26 October, 1999.